Find Domain

**Kiwigirl** · June 11th 2006, 06:44 PM

Can anyone please help me in equation form with the answers to the following questions?

1. Give the range of the function y = tan^-1 x

2. Give the value of tan^-1 5

**ThePerfectHacker** · June 11th 2006, 06:56 PM

Originally Posted by Kiwigirl

Can anyone please help me in equation form with the answers to the following questions?

1. Give the range of the function y = tan^-1 x

There are two ways to do this problem. One way is the cowardly way they teach in school using a graph in that case look below to see that the range is,
$y\in (-\pi/2,\pi/2)$

The second way. The function $\tan^{-1}(x)$ is defined as the inverse function of the function,
$\tan x \mbox{ on } -\pi/2<x<\pi/2$ . Now there is a rule that the range of the inverse function is the domain of the orignal function. Since the domain of the original function is $(-\pi/2,\pi/2)$ thus the range of $\tan^{-1}(x)$ is $(-\pi/2,\pi/2)$

There is actually a third way (much more advanced). Find all such $y$ such as the equation $y=\tan^{-1}(x)$ has a solution.

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